Wouldn't have worked, as I only had two happiness slots, which were filled. I ended up declaring martial law and then bought a happiness-increasing tech from the psi-fish. By the time I researched another tech and could build the structure it enables, I won the mission.Literally 15 minutes after posting, it occurred to me you could also manually reshuffle your colonists or set priorities to the automation. That allows you to put more colonists in "happiness" jobs, so you can actually avoid the martial law.
I'm still in the process of learning the game, which does include things like "but wait, where is the Volcano habitation tech for these guys" and how to manage expansion.